Question: Is the function f(x) = { sin(x), x < 0; x^2, x >= 0 } continuous at x = 0?
Options:
Yes
No
Depends on x
Not defined
Correct Answer: Yes
Solution:
Both limits as x approaches 0 from the left and right are equal to 0, hence f(x) is continuous at x = 0.
Is the function f(x) = { sin(x), x < 0; x^2, x >= 0 } continuous at x = 0?
Practice Questions
Q1
Is the function f(x) = { sin(x), x < 0; x^2, x >= 0 } continuous at x = 0?
Yes
No
Depends on x
Not defined
Questions & Step-by-Step Solutions
Is the function f(x) = { sin(x), x < 0; x^2, x >= 0 } continuous at x = 0?
Correct Answer: Yes, f(x) is continuous at x = 0.
Step 1: Identify the function f(x). It is defined as f(x) = sin(x) when x is less than 0 and f(x) = x^2 when x is greater than or equal to 0.
Step 2: Find the limit of f(x) as x approaches 0 from the left (x < 0). This means we will use the part of the function f(x) = sin(x). Calculate the limit: lim (x -> 0-) sin(x) = sin(0) = 0.
Step 3: Find the limit of f(x) as x approaches 0 from the right (x >= 0). This means we will use the part of the function f(x) = x^2. Calculate the limit: lim (x -> 0+) x^2 = 0^2 = 0.
Step 4: Compare the two limits from Step 2 and Step 3. Both limits are equal to 0.
Step 5: Check the value of the function at x = 0. Since f(0) = 0^2 = 0, the function value at x = 0 is also 0.
Step 6: Since the limit from the left (0), the limit from the right (0), and the function value at x = 0 (0) are all equal, we conclude that f(x) is continuous at x = 0.
Continuity of Piecewise Functions – Understanding how to determine the continuity of a function defined in pieces by evaluating limits from both sides and comparing them to the function's value at the point of interest.
Limit Evaluation – Calculating the left-hand limit and right-hand limit as x approaches a specific point to assess continuity.
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